Kapouleas and Yang have constructed, by gluing methods, sequences of
minimal embeddings in the round 3-sphere converging to the Clifford
torus counted with multiplicity 2. Each of their surfaces, which
they call doublings of the Clifford torus, resembles a pair of
coaxial tori connected by catenoidal tunnels and has symmetries
exchanging the two tori. I will describe an extension of their work
which yields doublings admitting no such symmetries as well as
examples incorporating an arbitrary (finite) number of tori, that
is Clifford torus triplings, quadruplings, and so on.

From a complex analytic perspective Teichmüller space - the universal
cover of the moduli space of Riemann surfaces - is a contractible
bounded domain in a complex vector space. Likewise, Bounded Symmetric
domains arise as the universal covers of locally symmetric varieties
(of non-compact type). In this talk we will study isometric maps
between these two important classes of bounded domains equipped with
their intrinsic Kobayashi metric.

We consider classes of diffeomorphisms of Euclidean space with partial asymptotic expansions at infinity; the remainder term lies in a weighted Sobolev space whose properties at infinity fit with the desired application. We show that two such classes of asymptotic diffeomorphisms form topological groups under composition. As such, they can be used in the study of fluid dynamics according to the method of V. Arnold. Specific applications have been obtained for the Camassa-Holm equation and the Euler equations.

I'll report on joint work with Nick Sheridan (Princeton/IAS) about mirror symmetry for Calabi-Yau (CY) manifolds. Kontsevich's homological mirror symmetry (HMS) conjecture proposes that the Fukaya category of a CY manifold (viewed as a symplectic manifold) is equivalent to the derived category of coherent sheaves on its mirror. We show that if one can prove an apparently weaker fragment of this conjecture, for some mirror pair, then one can deduce HMS for that pair. We expect this fragment  to be amenable to proof for the mirror pairs constructed in the Gross-Siebert program, for example.  We also show that the "closed-open string map" is an isomorphism, thereby opening a channel for proving the "closed string" predictions of mirror symmetry.